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<p>这里并不打算去详细推导KMP的原理，以及精髓。只是整理了一下KMP算法的模板、应用、理解的关键点。</p>
</blockquote>
<h1 id="基本功能"><a href="#基本功能" class="headerlink" title="基本功能"></a>基本功能</h1><p>KMP算法最基本的应用就是字符串的匹配——给定一个text串，一个pattern串，询问pattern串在text串中是否出现，以及出现几次、位置等等。</p>
<h1 id="模板"><a href="#模板" class="headerlink" title="模板"></a>模板</h1><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">class</span> <span class="title class_">KMP</span>&#123;</span><br><span class="line">    string pattern;</span><br><span class="line">    <span class="type">int</span> len ;</span><br><span class="line">    vector&lt;<span class="type">int</span>&gt; next;  <span class="comment">// 这里的next数组是从1开始的</span></span><br><span class="line"></span><br><span class="line">    <span class="comment">// 也叫失配数组,next[i] 表示以第i位结尾的子串与模式串的前缀相同的最大长度（但不能是自身），也就是所谓的最长公共前后缀</span></span><br><span class="line">    <span class="function"><span class="type">void</span> <span class="title">Next</span><span class="params">()</span></span>&#123;</span><br><span class="line">        next.<span class="built_in">resize</span>(len+<span class="number">1</span>);</span><br><span class="line">        <span class="type">int</span> j = <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">for</span>(<span class="type">int</span> i = <span class="number">2</span>; i &lt;= len; i++) &#123;</span><br><span class="line">            <span class="keyword">while</span>(j &gt; <span class="number">0</span> &amp;&amp; pattern[i] != pattern[j + <span class="number">1</span>]) &#123;</span><br><span class="line">                j = next[j];</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">if</span>(pattern[i] == pattern[j + <span class="number">1</span>]) &#123;</span><br><span class="line">                j++;</span><br><span class="line">            &#125;</span><br><span class="line">            next[i] = j;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span>:</span><br><span class="line">    <span class="built_in">KMP</span>(<span class="type">const</span> string &amp;pattern) &#123;</span><br><span class="line">        len = pattern.<span class="built_in">size</span>();</span><br><span class="line">        <span class="keyword">this</span>-&gt;pattern = <span class="string">&quot; &quot;</span> + pattern; </span><br><span class="line">        <span class="built_in">Next</span>();</span><br><span class="line">    &#125;</span><br><span class="line">    </span><br><span class="line">    <span class="function"><span class="type">bool</span> <span class="title">isSubstringOfText</span><span class="params">(<span class="type">const</span> string&amp; text)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> <span class="built_in">getPositions</span>(text).<span class="built_in">size</span>() &gt; <span class="number">0</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    </span><br><span class="line"></span><br><span class="line">    <span class="comment">// 统计模式串在文本串出现的次数</span></span><br><span class="line">    <span class="function"><span class="type">int</span> <span class="title">getTimes</span><span class="params">(<span class="type">const</span> string&amp; text)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> <span class="built_in">getPositions</span>(text).<span class="built_in">size</span>();</span><br><span class="line">    &#125;</span><br><span class="line">    </span><br><span class="line">    <span class="comment">// 统计模式串在文本串的出现的次数</span></span><br><span class="line">    <span class="function">vector&lt;<span class="type">int</span>&gt; <span class="title">getPositions</span><span class="params">(<span class="type">const</span> string&amp; text)</span> </span>&#123;</span><br><span class="line">        vector&lt;<span class="type">int</span>&gt; res;</span><br><span class="line">        <span class="keyword">if</span>(text.<span class="built_in">size</span>()  &lt; len) <span class="keyword">return</span> res;</span><br><span class="line">        <span class="type">int</span> j = <span class="number">0</span>;</span><br><span class="line">        <span class="keyword">for</span>(<span class="type">int</span> i = <span class="number">0</span>; i &lt; text.<span class="built_in">size</span>(); i++)&#123;</span><br><span class="line">            <span class="keyword">while</span> (j &gt; <span class="number">0</span> &amp;&amp; text[i] != pattern[j + <span class="number">1</span>])&#123;</span><br><span class="line">                j = next[j];</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">if</span>(text[i] == pattern[j + <span class="number">1</span>])&#123;</span><br><span class="line">                j++;</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">if</span>(j == len)&#123;</span><br><span class="line">                <span class="comment">// i 是匹配时文本串的位置（从0开始）</span></span><br><span class="line">                res.<span class="built_in">push_back</span>(i);</span><br><span class="line">                j = next[j];</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> res;</span><br><span class="line">    &#125;;</span><br><span class="line">    </span><br><span class="line"></span><br><span class="line">    <span class="comment">// 求数组f[] 其中 f[i] 表示文本串以i结尾的子串与 pattern 串的前缀相同的最大长度（这里也是从1开始计数）</span></span><br><span class="line">    <span class="function">vector&lt;<span class="type">int</span>&gt; <span class="title">getFArray</span><span class="params">(<span class="type">const</span> string&amp; text)</span> </span>&#123;</span><br><span class="line">        <span class="type">int</span> m = text.<span class="built_in">size</span>(), j = <span class="number">0</span>;</span><br><span class="line">        <span class="function">vector&lt;<span class="type">int</span>&gt; <span class="title">f</span><span class="params">(m+<span class="number">1</span>)</span></span>;</span><br><span class="line">        <span class="keyword">for</span>(<span class="type">int</span> i= <span class="number">0</span> ;i &lt; m; i++)&#123;</span><br><span class="line">            <span class="keyword">while</span>(j &gt; <span class="number">0</span> &amp;&amp; (j == len || text[i] != pattern[j + <span class="number">1</span>])) &#123;</span><br><span class="line">                j = next[j];</span><br><span class="line">            &#125;</span><br><span class="line">            <span class="keyword">if</span>(text[i] == pattern[j + <span class="number">1</span>])&#123;</span><br><span class="line">                j++;</span><br><span class="line">            &#125;</span><br><span class="line">            f[i] = j;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> f;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="type">const</span> vector&lt;<span class="type">int</span>&gt; &amp;<span class="title">getNext</span><span class="params">()</span> <span class="type">const</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> next;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;;</span><br></pre></td></tr></table></figure>

<h1 id="知识点"><a href="#知识点" class="headerlink" title="知识点"></a>知识点</h1><ul>
<li>失配数组的求法以及含义。<code>next[]</code></li>
<li>模式匹配的结果数组。<code>f[]</code></li>
<li>从DFA(有限状态自动)的角度去理解。(对后面的AC自动机的理解大有裨益)</li>
</ul>
<h1 id="典型应用"><a href="#典型应用" class="headerlink" title="典型应用"></a>典型应用</h1><ul>
<li>模式串和文本串的模式匹配。</li>
<li>处理和前缀和后缀有关的问题。</li>
<li>处理循环节方面的问题。</li>
</ul>

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